Intertwining numbers; the three-rowed case
Suppose that \(\lambda\) is a partition, \(\lambda=(\lambda_1,\dots,\lambda_n)\), \(d\) is a positive integer, and \(\mu\) is the partition obtained from \(\lambda\) by taking \(d\) boxes away from some row of \(\lambda\) and attaching them to some higher row (assuming that we still obtain a partition), i.e., \(\mu=(\lambda_1,\dots,\lambda_k+d,\dots,\lambda_{k+j}-d,\dots,\lambda_n)\). A standard question in modular representation theory is the following: What is the \(\mathbb{Z}/(p)\)-dimension of the \(\mathbb{Z}/(p)\)-vector space \(\text{Ext}^i_{A_r}(\overline K_\lambda,\overline K_\mu)\)? (The bars over the modules indicate the reduction of these modules modulo \(p\).) These numbers are called intertwining numbers. The attempts to determine these intertwining numbers are varied. One approach is suggested by the following special case of a theorem found in [\textit{K. Akin} and \textit{D. A. Buchsbaum}, Adv. Math. 72, No. 2, 171-210 (1988; Zbl 0681.20028)]. Let \(M\) and \(N\) be integral polynomial representations of \(\text{Gl}(F)\) of degree \(r\), let \(p\) be a prime, and let \(A_r\) be the integral Schur algebra of degree \(r\). Denote by \(\overline X\) the \(\mathbb{Z}/(p)\)-module (-algebra) \(X\otimes_\mathbb{Z}\mathbb{Z}/(p)\). Then we have the exact sequence \[ 0\to\text{Ext}^i_{A_r}(M,N)\otimes\mathbb{Z}/(p)\to\text{Ext}^i_{\overline A_r}(\overline M,\overline N)\to\text{Tor}^1_\mathbb{Z}(\text{Ext}^{i+1}_{A_r}(M,N),\mathbb{Z}/(p))\to 0. \] From this exact sequence, we see that it suffices to calculate the integral Ext groups, since the modular ones are simply the \(p\)-torsion part of one integral Ext plus the reduction modulo \(p\) of another. A fairly straightforward argument shows that it is enough to consider the case where we take \(d\) boxes from the last row and attach them to the first one. Of course, straight homological algebra tells us that we merely have to find a projective resolution, \(\mathbb{P}\), of \(K_\lambda\) over \(A_r\), and then simply calculate the cohomology of \(\text{Hom}_{A_r}(\mathbb{P},K_\mu)\). (The notation \(K_\lambda\) indicates the Weyl module associated to the partition \(\lambda\).) In [loc. cit.], it was proven that \(K_\lambda\) admits a finite projective resolution whose terms are direct sums of tensor products of divided powers (suitable tensor products of divided powers are \(A_r\)-projective). Moreover, it was shown that, if \(\alpha=(\alpha_1,\dots,\alpha_n)\) is an integral weight of degree \(r\), then \(\text{Hom}_{A_r}(D_{\alpha_1}\otimes\cdots\otimes D_{\alpha_n},N)\) is the weight submodule of the representation \(N\) corresponding to the weight \(\alpha\). Therefore, in our case, with \(N=K_\mu\), the calculation of \(\text{Hom}_{A_r}(\mathbb{P},K_\mu)\) comes down to the calculation of certain weight submodules of \(K_\mu\). That this is computable is due to the fact that the weight submodule of \(K_\mu\) is the free abelian group generated by the standard tableaux of shape \(\mu\) and content \(\alpha\). Now we know that \(\text{Hom}_{A_r}(\mathbb{P},K_\mu)\) is a complex of free abelian groups, and that its cohomology is torsion (since over the rationals the cohomology is zero). If we write down the maps in \(\text{Hom}_{A_r}(\mathbb{P},K_\mu)\) as integral matrices, elementary arguments show us that the cohomology groups are determined by the invariant factors of those matrices (the non-zero invariant factors). To calculate those matrices, we must know the maps in the projective resolution \(\mathbb{P}\). The problem of exhibiting such resolutions with their maps is still very much an ongoing one. However, we can momentarily avoid the problem of exhibiting these resolutions if we are willing to study just the intertwining numbers for \(i=0\), that is, if we just want to study the dimension of \(\text{Hom}_{\overline A_r}(\overline K_\lambda,\overline K_\mu)\). For in this case, as already observed, we need only calculate the \(p\)-torsion of \(\text{Ext}^1_{A_r}(K_\lambda,K_\mu)\) (the corresponding Hom group is zero), and from the remarks above about the nature of the cohomology of \(\text{Hom}_{A_r}(\mathbb{P},K_\mu)\), it suffices to know a presentation of \(K_\lambda\). And this we do know, thanks to the proof of the standard basis theorem. For two-rowed shapes and arbitrary \(d\), the problem was solved many years ago by \textit{K. Akin} and \textit{D. A. Buchsbaum} [in Commutative algebra, Berkeley 1989, Publ., Math. Sci. Res. Inst. 15, 1-19 (1989; Zbl 0744.20037)]. In that same article, the integral matrix whose invariant factors have to be computed was written down explicitly. It was conjectured, but not yet proven, that the Ext group was cyclic. In 1991, \textit{D. Flores de Chela} [J. Algebra 171, No. 2, 631-653 (1995; Zbl 0841.16017)] proved that it was indeed cyclic and we then started to look for the highest invariant factor of the integral matrix. This article describes the invariant factor, and proves the validity of the description.
scientific article; zbMATH DE number 923700